This document describes the method of regularized Stokeslets for modeling Stokes flow. It introduces Stokes flow equations and the Green's function solution of a Stokeslet. It then describes how the method of regularized Stokeslets assumes forces are spread over a small region rather than a Dirac delta function. It provides examples of regularized shape functions in 2D and 3D and derives the corresponding regularized Green's functions Gε and Bε. The document concludes by describing an algorithm to solve for forces given velocities and provides examples comparing results to analytical solutions for 2D and 3D flows.

1. THE METHOD OF REGULARIZED STOKESLETS
Mingjie Zhu
14th, August, 2015
Beijing Computational Science Research Center

2. STOKES FLOW
In the common case of an incompressible Newtonian fluid, the Stokes equations
take the (vectorized) form:
𝜇𝛻2
− 𝛻𝑝 + 𝐅 = 𝟎
𝛻 ∙ 𝒖 = 𝟎

3. GREEN'S FUNCTION SOLUTION: THE STOKESLET
We first consider the generic situation in which the force are spread over a small ball
centered at the points 𝒙0. The force is given by
𝑭 𝒙 = 𝒇0 𝛿(𝒙 − 𝒙0)
Then we define a function G(𝑥)called Green’s function
𝛻2G 𝑥 = 𝛿(𝑥)
And another function
𝛻2
B 𝑥 = G 𝑥

4. GREEN'S FUNCTION SOLUTION: THE STOKESLET
Take the divergence of Stokes equations, we find that the pressure satisfies
𝛻2
𝑝 = 𝛻 ∙ 𝐅
Which gives the particular solution
𝑝 = 𝒇0 ∙ 𝛻G
Now we use this expression to rewrite the equation for 𝒖 as 𝜇𝛻2
𝒖 = 𝒇0 ∙ 𝛻 𝛻G − 𝒇0 𝛿,
whose particular solution is
𝜇𝒖 𝑥 = 𝒇0 ∙ 𝛻 𝛻B 𝒙 − 𝒙0 − 𝒇0G 𝒙 − 𝒙0

5. GREEN'S FUNCTION SOLUTION: THE STOKESLET

6. REGULARIZED STOKESLETS
The method of regularized Stokeslets assumes the force is not a Dirac delta applied at a
single point but uses a shape function. We use 𝜙 𝜖(𝒙) to denote this function, which is a
radially symmetric smooth function with the property that 𝜙 𝜖(𝒙) d𝒙 = 1. The
parameter 𝜖 determines the spread of the function.

7. EXAMPLES OF THESE FUNCTIONS
In 𝐑2: 𝜙 𝜖 𝒙 =
3𝜖3
2𝜋( 𝒙 2+𝜖2)5/2
In 𝐑3
: 𝜙 𝜖 𝒓 =
15𝜖4
8𝜋(𝒓2+𝜖2)7/2

8. DERIVING G 𝜖 AND B 𝜖
Since 𝜙 𝜖 is radially symmetric and 𝛻2
G 𝜖 = 𝜙 𝜖, we have that
𝛻2
G 𝜖 =
1
𝑟
𝑟G 𝜖
′
𝑟 ′
= 𝜙 𝜖(𝑟)
And
G 𝜖
′
𝑟 =
1
𝑟 0
𝑟
𝑠 𝜙 𝜖 𝑠 d𝑠
After integrating once more, we obtain G 𝜖(𝑟). Then, we can get B 𝜖(𝑟) in the same way.

9. FINAL RESULTS

10. FINDING FORCES FROM VELOCITIES
We can write last equations into
𝑢 𝒙𝑖 =
𝑗=1
𝑁
𝑀𝑖𝑗(𝒙1, 𝒙2, … , 𝒙 𝑁) 𝒇 𝑗
Or, as a matrix equation:
𝑼 = 𝑴𝑭
Which is to solve F using U and F. However, it is
impossible to calculate the inverse of 𝑴
directly. We use GMRES with zero initial guess.
Radius=1, velocity=(0.4, 0)

11. ALGORITHM
1. Create all Stokeslets: define their coordinates and initial velocities;
2. Calculate each element in the matrix 𝑴;
3. Solve the matrix equation using GMRES to obtain forces in each Stokeslet; (Normally,
GMRES can iterate converging to the tolerance 1e-8)
4. Calculate velocity of other points in the Stokes flow field;
5. Plot the result in the screen or output these result.

12. RESULTS COMPARE: A 2D CIRCLE
Radius=0.25, velocity=(1, 0)
Up: the x component of velocity
Down: the y component of velocity

13. RESULTS COMPARE: A 3D SPHERE
Radius=1.2, velocity=(0.5, 0, 0)
When z=0.
Up: the x component of velocity
Down: the y component of velocity

14. RESULTS COMPARE: A 3D SPHERE
x y stokes_u stokes_v analytical_u analytical_v error_u error_v
1 1 0.346046 0.034137 0.351291 0.0356382 0.014931 0.042133
1 1.3 0.29273 0.04787 0.296579 0.0492919 0.012979 0.028851
1 1.6 0.245579 0.049912 0.248442 0.0510662 0.011525 0.022603
1 1.9 0.207925 0.046623 0.210136 0.0475185 0.010524 0.018845
1 2.2 0.178654 0.041589 0.180432 0.042281 0.009856 0.016366
1 2.5 0.155852 0.036406 0.157331 0.0369465 0.009398 0.014642
1 3.7 0.101882 0.021099 0.102764 0.0213386 0.008585 0.011207
1 4 0.093677 0.018602 0.0944791 0.018804 0.008491 0.010768
1.6 4 0.095528 0.026299 0.0963557 0.0265792 0.008586 0.010535
1.6 4.3 0.088386 0.023647 0.0891441 0.0238912 0.0085 0.010234
1.6 4.6 0.082207 0.021328 0.0829062 0.0215431 0.008432 0.009984
1.6 4.9 0.076817 0.0193 0.0774665 0.0194909 0.008378 0.009774
1.9 1 0.27197 0.046623 0.275412 0.0475185 0.012498 0.018845
1.9 2.2 0.173029 0.050128 0.174771 0.0508224 0.009968 0.013657

15. RESULTS COMPARE: A 3D SPHERE

16. APPLICATION: BACTERIA

17. IMPLEMENTATION USING C++
1. Run faster (Native code and OpenMP supported);
2. Better readability;
3. Expandable.